Let $(\overline{M},\overline{g}) = (\mathbb{R}^3,g_{\text{eucl}})$ and let $(M,g)$ be a semi-riemannian hypersurface. Let $p \in M$ and $\xi$ a unit normal field in a neighbourhood $U \subset M$ of $p$ and let $\{e_1,e_2\}$ be an orthonormal basis for $T_pM$. Now, we have the formula $$0 = \overline{K}(e_1,e_2) = K(e_1,e_2)-\overline{g}(\Pi(e_1,e_1),\Pi(e_2,e_2))+\overline{g}(\Pi(e_1,e_2),\Pi(e_1,e_2))$$ which will reduce to $$K(e_1,e_2) = g(S^{\xi}(e_1),e_1)\cdot g(S^{\xi}(e_2),e_2)-g(S^{\xi}(e_1),e_2)^2$$ $$\iff$$ $$K(e_1,e_2) = \operatorname{det}(S^{\xi}).$$
Now, I tried this formula to calculate the gauß curvature of the torus in $\mathbb{R}^3$, and it seems like I get the right answer if I in the last calculation use the euclidian metric instead of the metric for the torus, i.e. $$g_{\mathbb{T}^2} = (R+r\cos(\varphi))^2d\theta^2+r^2d\varphi^2.$$ But it seems from this formula like I actually should use the metric $g = g_{\mathbb{T}^2}$ on $M = \mathbb{T}^2$, the torus, in the last part of the calculation. What am I missing? So in this particular example with the torus, should I not do the identification $$(M,g) = (\mathbb{T}^2,g_{\mathbb{T}^2})?$$
This issue has been resolved. I missed that with the metric from the torus $$g_{\mathbb{T}^2} = (R+r\cos(\varphi))^2d\theta^2+r^2d\varphi^2$$ and a point $p \in \mathbb{T}^2$, $$g_{\mathbb{T}^2}(\partial_{\theta}|_p,\partial_{\theta}|_p)\cdot g_{\mathbb{T}^2}(\partial_{\varphi}|_p,\partial_{\varphi}|_p)-g_{\mathbb{T}^2}(\partial_{\theta}|_p,\partial_{\varphi}|_p)^2$$ is not equal to $1$, in the denominator of one of the terms in the expression $$ 0 = \overline{K}(\partial_{\theta}|_p,\partial_{\theta}|_p)$$$$ = K(\partial_{\theta}|_p,\partial_{\theta}|_p) - \frac{\overline{g}(\Pi(\partial_{\theta}|_p\partial_{\theta}|_p),\Pi(\partial_{\varphi}|_p,\partial_{\varphi}|_p)) - \overline{g}(\Pi(\partial_{\theta}|_p,\partial_{\varphi}|_p),\Pi(\partial_{\theta}|_p,\partial_{\varphi}|_p)}{g_{\mathbb{T}^2}(\partial_{\theta}|_p,\partial_{\theta}|_p) \cdot g_{\mathbb{T}^2}(\partial_{\varphi}|_p,\partial_{\varphi}|_p) - g_{\mathbb{T}^2}(\partial_{\theta}|_p,\partial_{\varphi}|_p)^2} $$
where $\overline{g}$ is the euclidian metric on $\mathbb{R}^3$.